Secret Sharing Schemes for Graph-Based Prohibited Structures

K e y w o r d s C r y p t o g r a p h y , Data security, Information theory, Secret sharing scheme. 1 . I N T R O D U C T I O N In 1987, I to et al. [1] desc r ibed a genera l m e t h o d of secret shar ing cal led Secret Sharing Scheme (SSS) which allows a m a s t e r key to be sha red among a finite set of p a r t i c i p a n t s in such a way t h a t on ly ce r ta in prespecif ied subse ts of p a r t i c i p a n t s can recover the mas t e r key. Let P be t he set of pa r t i c ipan t s . T h e set of all subse ts of P , deno ted by 2 P, is cal led the power set of P . We use the n o t a t i o n X \ Y = {x I x E X and x ~ Y} to denote t he difference of two sets X and Y. T h e col lect ion of subse t s of pa r t i c i pan t s t h a t can recons t ruc t t he m a s t e r key in th i s way is cal led access structure (deno ted by F) . T h e col lect ion of subse ts of p a r t i c i p a n t s t h a t canno t r econs t ruc t the mas t e r key is ca l led prohibited structure (denoted by A ) [2]. T h e n a t u r a l r es t r i c t ion is t h a t r is m o n o t o n e increas ing and A is mono tone decreasing, t h a t is, if A E F and A C_ B C P , if A E A and B C_ A C_ P , then B ~ F,